Lee–Lerario–Lundberg zero-count bound for harmonic polynomial equations

Let $p(z)$ and $q(z)$ be polynomials satisfying

$${\rm \deg}(p(z))=n,\qquad {\rm \deg}(q(z))=m,$$

with $n>m$. Consider the equation

$$p(z)=\overline{q(z)}.$$

Lee–Lerario–Lundberg bound. The number of solutions of this equation is bounded by

$$2m(n-1)+n.$$

This bound is proposed after counterexamples invalidate Wilmshurst's earlier conjectured bound. The source presents it as a proposed replacement, rather than as a proved theorem.